Hacker News Reader

Comment by alexmolas | Hacker News Reader

But you can also think of any real number, such as pi or e, which came from an uncountable set.

You can't think of any real number. You can think of some of them, but you need to have a way of thinking of them with some specification that can be written down, and those form a countable set. We also have finite life spans, so we actually can only think of finitely many numbers. When you think of the natural numbers, you think of them as a set and concept, which is different than being able to visualize and conceive of every natural number.

What about potentially think of? Even that does not make sense because every possible description you can think of must be written with finitely many symbols in some language, and all such descriptions form a countable set.

Therefore, there are real numbers that could never be written down in English, even potentially. Of course, that partially also depends on the kind of axiom system that you are using. In effect, the standard axioms of mathematics state the existence of things that cannot be effectively specified, which some people actually are against, although such people form a minority.

The surprising thing is that we can effectively reason with large infinities and such objects make intuitive sense (the set of all functions from the natural numbers to the natural numbers is uncountable but a very natural sounding set), and yet it is impossible even in principle to write every one down, even with an unlimited amount of time.

Real numbers do not make intuitive sense. The continuum makes some sense. A countably subset of the reals makes sense. The rest of the reals (almost all of them) are beyond comprehension, except in vague description like "an infinite string of digits" and for proofing paradoxes like Banach-Tarski.

Pi and E can be thought of because there are mathematical formulas and algorithms relating them and computing them. There are only countably many such formulas & algorithms.

You can't think of an arbitrary real number, in the sense of distinguishing it every other real number(including nearby reals an ε away from it), and with no other constraints besides being a real number.

it's simple, we'll count the number of epsilons we need away from the nearest described number... wait...

> But you can also think of any real number

I don't think that's true. We can certainly think of certain numbers which we gave a name to and have defined it in some way. But there are a _lot_ of real numbers.

We could think about it this way: we can only describe (and therefore think of) numbers using a finite number of symbols out of a finite alphabet. That makes it only countable.

Can we think of any computable number?

And then some uncomputable ones if we give them a name/concept.

It seems quite wild

What does it take to think of a number? We can certainly refer to particular uncomputable numbers - choose an enumeration of Turing machines, and let x be the real number whose binary expansion has 1 at the i-th place if the i-th Turing machine halts, and 0 otherwise.

There are computable numbers whose minimum expression requires more bits than the universe can hold, using any possible encoding the universe has sufficient bits to define.

Infinity is much bigger than any finite number. This includes even "all possible numbers reachable in all possible finite encodings of a number within a given size", as inconceivably large as that is. No matter how large that set is, somewhere, there is some maximum number that is the largest possible number you can specify under the conditions I've given here, and that number is 0% of the way along the number line, which means that 100% of all "computable numbers" are larger than than that number.

There are some uncomputable numbers that have been given names. We even have the first few digits of some of them. See for instance: https://en.wikipedia.org/wiki/Chaitin%27s_constant

But, unless you have predefined a language, you can't know which countable set is expressible, so the union of all possible sets may be uncountable.

But I think all languages is countable, so it should be a subset of the product of two countable sets, which should be countable?

Pi and e come from an uncountable set, but the set of things you can think of is almost certainly countable.

Reply on news.ycombinator.com